Lifeworld and Mathematics Rudolf Wille Fachbereich Mathematik, Technische Universit¨atDarmstadt, Schloßgartenstr. 7, D{64289 Darmstadt [email protected] Abstract. The article \Lifeworld and Mathematics" has been inspired by well-known scientists, from whom are listed here: Edmund Husserl, J¨urgenHabermas, Reuben Hersh, Martin Heidegger, Hartmut von Hentig and Knut Radbruch. Basic for this article is Husserl's phenomenological lifeworld analysis of the mathematized modern natural science. Haber- mas, in whose social theory the concept of lifeworld has also a central meaning, recommends a theoretic communicational approach for the life- world analysis which answers the question about the intersubjective con- stitution of the lifeworld in the sense of the american pragmatism. Hersh has a pragmatic understanding of mathematics so that he views math- ematics as a social, cultural, historical reality. Heidegger saw the basic character of the modern knowledge attitude in the new knowledge claim named the \mathematical" which is not deducable out of mathematics. Hentig has offered the book \Magier oder Magister? Uber¨ die Einheit der Wissenschaft im Verst¨andigungsprozess",in which he discusses gen- erally the role of the science in our society, while doing so he considers as necessary a democratically activated general understanding of scientific actions. Already 1991 Radbruch has pointed out in a lecture at the Darm- stadt Seminar about \general mathematics" the analysis of conditions and realizations of the mathematical as task of general mathematics. 1 Lifeworld and Inworld The concept \lifeworld" has been introduced into philosophy by Edmund Husserl in his late work \The crisis of the european sciences and the transcendental phenomenology" (see [Hu54]). With the term \lifeworld" Husserl names - in the frame of his phenomenological philosophy - the concrete context of the world. This context is intersubjectively experienced by humans in original evidence and is therefore ordered before the objective-scientific cognition of the world. Husserl founded on the \lifeworld" concept his fundamental critic about the development of science. For him the scientists have chanced the ideal of objectivity, which has to be understood only methodically, into an independently made \objectivism" by which the relationship of the research to the responsibility of the acting human being is lost. To overcome by it the occuring sense-crisis of science, Husserl recommends to make conscious that and how the objectivistic imagined world Lifeworld and Mathematics 23 arises out of human achievements which are founded on the lifeworld as the extensive horizon of human cognition and action. Husserl's phenomenological lifeworld analysis of the mathematized modern natural science has been extended by Alfred Sch¨utzwith a phenomenological analysis of the social lifeworld [SL79]. Sch¨utzunderstands the lifeworld as the transcendental frame of possible everyday experiences of the recognizing and acting subject. J¨urgenHabermas, in whose social theory the concept of lifeworld has also a central meaning, criticizes the approach of Sch¨utz:On the one hand Sch¨utztends to start out from an intersubjectively constituted lifeworld, without to clarify how the lifeworld is intersubjectively produced (a problem of which Sch¨utzknew that also Husserl has not solved it); on the other hand he takes over Husserl's philosophical view of consciousness to comprehend the \experiencing subject" as the last reference point of the lifeworld analysis [Ha81; vol.2, p.197f]. Habermas recommends a theoretic communicational approach for the lifeworld analysis which answers the question about the intersubjective constitution of the lifeworld in the sense of the american pragmatism. A convincing understanding of mathematics has also to be pragmatically founded, as it is for instance realized by Reuben Hersh in his 1997 appeared book \What is mathematics, really?" [He97]. Hence, the treatment of the theme \lifeworld and mathematics" shall be based in this paper on Habermas' concept of lifeworld, which has therefore be more explicated in detail. J¨urgenHabermas introduces in his main work \Theorie des kommunikativen Handelns" [Ha81] \lifeworld" as a complementary concept to the \communica- tive action". Habermas understands action and therefore communicative action as mastering of situations. \The concept of communicative action cuts out of the mastering of situations first of all two aspects: the teleological aspect of the realization of the mastering of purposes (or the execution of a plan of action) and the communicative aspect of the interpretation of a situation and the achiev- ing of an agreement. In the communicative action the participants pursue their plans on the basis of common definitions of situations in mutual agreements." [Ha81; vol.2, p.193] For the unification of common definitions of situations the lifeworld is fundamental because \communicatively acting subjects communi- cate always in the horizon of a lifeworld. [...] This lifeworld background serves as a source for defining situations which are presupposed by the participants as unproblematic." [Ha81, vo.1, p.107] For the situation-oriented communication \the lifeworld is a reservoir of self-evident facts or unshakable convictions which the communication participants utilize for interpretation processes. But single elements and determined self-evident facts are only mobilized in the form of consentaneous and at the same time of problematic knowledge if they become relevant for some situation. [...Out of an understanding-oriented view] we are able to think the lifeworld as a culturally transmitted and linguistically orga- nized supply of explanation patterns." [Ha81; vol.2, p.189] To be able in answering the question about the intersubjective constitution and reproduction of the lifeworld, Habermas goes back to the social psychol- ogy of the american pragmatist Georg Herbert Mead. Mead, who is interested 24 Rudolf Wille in the complementary structure of the subjective and social world, desribes in his book \Mind, Self, and Society" how speech becomes a medium of socializa- tion and social integration and how personal identities and social institutions develop through liguistic arrangements and normative steering interactions. So- cialization and social integration, which produce symbolic structures of the self and the society and with it competences and patterns of relationships, are exe- cuted by acts of communication; however Mead does not analyses thematically that these processes of communications are also be reflected in intersubjectively constituted cultural knowledge. To make it more clear that the speech mediated normative interaction - besides subjective competences and social patterns of relations - also leads to a propositional differenciated linguistic communication, Habermas distinguishes more clearly as Mead \between speech as a medium of communication and speech as a medium of the coordination of actions and of the incorporation of individuals." [Ha81; vol.2, p.41] This treefold function takes over the speech in the communicative action: \Under the functional aspect of communication, the communicative action serves the tradition and renewing of cultural knowledge; under the aspect of the coordi- nation of action, it serves the social integration and the production of solidarity; finally, under the aspect of socialization, the communicative action serves the instruction of personal identities." [Ha81; vol.2, p.208] In the communicative ac- tion the constitution and reproduction of the lifeworld take place and to be more precise by renewing and continuing of valid knowledge, through develpopment and stabilisation of legitimate orders and of solidarity of groups as well as by the training of sound of mind and competent acting persons. Habermas goes with his lifeworld understanding beyond the area of knowledge because \the sol- idarities of the about values and norms integrated groups and the competences of incorperated individuals in simular manner as cultural tradition [...] flow in communicative action." [Ha81; vol.2, p.205] As a reference system for describing and explaning, which concerns a life- world as a whole and do not only regard occurrences, Habermas turned out culture, society, and person as the structural components of the lifeworld which corresponds the proceedings of the cultural reproduction, the social integration, and the socialization. What Habermas wants to understand in this connection by culture, society, and person, he lays down as follows: \I entitle \culture" as the knowledge supply out of which the communication partners provide them- selves with interpretations to come to an agreement about something in the world. I entitle \society" as the legitimate order about which the communica- tion participants organize their membership in social groups and secure with it their solidarity. I understand under \Personality" the competences which makes a subject to have speech and capacity to act, i.e. to renovate, to participate in communications and at the same time to declare the own identity." [Ha81; vol.2, p.209] To make clear how the lifeworld differentiates itself in its symbolic structures, it recommends to describe in detail the three processes of reproduction and their connections: Lifeworld and Mathematics 25 • \The cultural reproduction of the lifeworld [...] secures the continuity of the tradition and a sufficient coherence of knowledge for the everyday practice, re- spectively.